Question

CRP-1819-JEEA-RP-PI-MA-4 dx is equal to The value of $10* + 1/2e* +3) 19 (A) x + in(e* + 1) – žin (2e* +3)+C (C) * -in (e* + 1) + In (20* + 3) +C - (B) x - In(e* + 1) + 3in (20* + (D) none of these IN

JEE/Engineering Exams

Maths

Solution

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( 0 quad int frac{d x}{left(e^{x}+1right)left(2 e^{x}+3right)} ) 0 ( sqrt{frac{000000}{10000}} ) Let ( e^{x}=t Rightarrow x=log t ) diffenciating both sides: ( frac{d x}{d x}=frac{d log t}{d x} Rightarrow d x=frac{1}{t} d t ) fum 1 i: ( begin{aligned} int frac{d t}{t(t+1)(2 t+3)} & text { (setring by partial fraction) } end{aligned} ) wh ( frac{1}{t(t+1)(2 t+3)}=frac{A}{t}+frac{B}{t+1}+frac{c}{2 t+3} ) ( =Aleft(2 t^{2}+5 t+3right)+Bleft(2 t^{2}+3 tright)+cleft(t^{2}+tright) ) On Conuparing like ferns both sides: ( 2 A+2 B+C=0 ) ( S A+3 B+C=0 ) ( 3 A=1 quad Rightarrow A=1 / 3 ) ( frac{1}{t(1+t)(2 t+3)}=frac{1}{3 t}-frac{1}{(1+t)}+frac{4}{3(2 t+3)} ) Futegenting boll sides 2 ( int frac{d t}{t(1+t)(2 t+3)}=int frac{d t}{3 t}-int frac{d t}{1+t}+frac{4}{3} int frac{d t}{2 t+3} ) ( =frac{1}{3} log t-log (1+t)+frac{2}{3} log (2 t+3) ) ( =frac{1}{3} x-log left(1+e^{x}right)+frac{2}{3} log left(2 e^{x}+3right)+6 )